Condition on two random variables

Question

I'm trying to set up the proper assumptions for a proof I'm working on:

Given that $P(A|e) = P(A)$ and $P(A|c,e) = P(A|e)$, can we prove that $P(A|c)=P(A)$?

I understand that A is independent of e and A is also independent of c given e, thus we can say A is independent of c.

I think this is relatively trivial to prove but I can't just give it out. Do I miss any assumptions to make it work?

Answer 1

The statement is true. You can use the fact the fact that you can obtain a marginal probability by taking the expectation of a conditional probability, i.e. \begin{align*} \mathbb{P}(A \mid C=c) &= \mathbb{E}[\mathbb{P}(A \mid C =c,E) \mid C=c] \\ &= \mathbb{E}[P(A) \mid C =c ] \\ &= P(A). \end{align*} This is the breakdown of each equality

• The first equality holds by integrating the conditional probability. To see the intuition for why, suppose that $E$ is discrete $$ \mathbb{P}(A \mid C =c) = \sum_e \mathbb{P}( A \mid C = c,E=e)\mathbb{P}(E = e \mid C = c) $$
• The second equality substitutes the relationship in your problem.
• The third equality uses the fact that $\mathbb{P}(A)$ is a constant.